Upper central series members are completely divisibility-closed in nilpotent group

From Groupprops
Jump to: navigation, search

Statement

Suppose G is a nilpotent group. Then, the members of the upper central series of G (with the possible exception of the zeroth member, the trivial subgroup) are completely divisibility-closed subgroups of G. Explicitly, this means that if p is a prime number such that G is p-divisible, then all the members of the upper central series of G (possibly excepting the zeroth member, the trivial subgroup) are completely p-divisibility-closed subgroups of G.

In particular, the center and second center are both completely divisibility-closed subgroups, and hence also divisibility-closed subgroups.

Related facts

Similar facts

Opposite facts

Facts similar to proof technique

Facts used

  1. Annihilator of divisibility-closed subgroup under bihomomorphism is completely divisibility-closed
  2. Complete divisibility-closedness is strongly intersection-closed
  3. Complete divisibility-closedness is transitive

Proof

The proof of this is unusual in that the induction proceeds downward from the penultimate member (the largest). The idea is to shift the "take the p^{th} root" operation between the elements in the iterated commutator map and note that the output is unaffected. This forces all p^{th} roots of an element in the appropriate upper central series member to be in that upper central series member.